3.13.97 \(\int (a+b x) (c+d x)^{5/2} \, dx\)

Optimal. Leaf size=42 \[ \frac {2 b (c+d x)^{9/2}}{9 d^2}-\frac {2 (c+d x)^{7/2} (b c-a d)}{7 d^2} \]

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Rubi [A]  time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {43} \begin {gather*} \frac {2 b (c+d x)^{9/2}}{9 d^2}-\frac {2 (c+d x)^{7/2} (b c-a d)}{7 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)*(c + d*x)^(5/2),x]

[Out]

(-2*(b*c - a*d)*(c + d*x)^(7/2))/(7*d^2) + (2*b*(c + d*x)^(9/2))/(9*d^2)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int (a+b x) (c+d x)^{5/2} \, dx &=\int \left (\frac {(-b c+a d) (c+d x)^{5/2}}{d}+\frac {b (c+d x)^{7/2}}{d}\right ) \, dx\\ &=-\frac {2 (b c-a d) (c+d x)^{7/2}}{7 d^2}+\frac {2 b (c+d x)^{9/2}}{9 d^2}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 30, normalized size = 0.71 \begin {gather*} \frac {2 (c+d x)^{7/2} (9 a d-2 b c+7 b d x)}{63 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)*(c + d*x)^(5/2),x]

[Out]

(2*(c + d*x)^(7/2)*(-2*b*c + 9*a*d + 7*b*d*x))/(63*d^2)

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IntegrateAlgebraic [A]  time = 0.03, size = 33, normalized size = 0.79 \begin {gather*} \frac {2 (c+d x)^{7/2} (9 a d+7 b (c+d x)-9 b c)}{63 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)*(c + d*x)^(5/2),x]

[Out]

(2*(c + d*x)^(7/2)*(-9*b*c + 9*a*d + 7*b*(c + d*x)))/(63*d^2)

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fricas [B]  time = 1.22, size = 93, normalized size = 2.21 \begin {gather*} \frac {2 \, {\left (7 \, b d^{4} x^{4} - 2 \, b c^{4} + 9 \, a c^{3} d + {\left (19 \, b c d^{3} + 9 \, a d^{4}\right )} x^{3} + 3 \, {\left (5 \, b c^{2} d^{2} + 9 \, a c d^{3}\right )} x^{2} + {\left (b c^{3} d + 27 \, a c^{2} d^{2}\right )} x\right )} \sqrt {d x + c}}{63 \, d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c)^(5/2),x, algorithm="fricas")

[Out]

2/63*(7*b*d^4*x^4 - 2*b*c^4 + 9*a*c^3*d + (19*b*c*d^3 + 9*a*d^4)*x^3 + 3*(5*b*c^2*d^2 + 9*a*c*d^3)*x^2 + (b*c^
3*d + 27*a*c^2*d^2)*x)*sqrt(d*x + c)/d^2

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giac [B]  time = 1.61, size = 306, normalized size = 7.29 \begin {gather*} \frac {2 \, {\left (315 \, \sqrt {d x + c} a c^{3} + 315 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a c^{2} + \frac {105 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} b c^{3}}{d} + 63 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} a c + \frac {63 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} b c^{2}}{d} + 9 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} a + \frac {27 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} b c}{d} + \frac {{\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} - 180 \, {\left (d x + c\right )}^{\frac {7}{2}} c + 378 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{2} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{3} + 315 \, \sqrt {d x + c} c^{4}\right )} b}{d}\right )}}{315 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c)^(5/2),x, algorithm="giac")

[Out]

2/315*(315*sqrt(d*x + c)*a*c^3 + 315*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a*c^2 + 105*((d*x + c)^(3/2) - 3*sq
rt(d*x + c)*c)*b*c^3/d + 63*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*a*c + 63*(3*(d*x
+ c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*b*c^2/d + 9*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2)
*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*a + 27*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2)*c + 35*(d*x
+ c)^(3/2)*c^2 - 35*sqrt(d*x + c)*c^3)*b*c/d + (35*(d*x + c)^(9/2) - 180*(d*x + c)^(7/2)*c + 378*(d*x + c)^(5
/2)*c^2 - 420*(d*x + c)^(3/2)*c^3 + 315*sqrt(d*x + c)*c^4)*b/d)/d

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maple [A]  time = 0.00, size = 27, normalized size = 0.64 \begin {gather*} \frac {2 \left (d x +c \right )^{\frac {7}{2}} \left (7 b d x +9 a d -2 b c \right )}{63 d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)*(d*x+c)^(5/2),x)

[Out]

2/63*(d*x+c)^(7/2)*(7*b*d*x+9*a*d-2*b*c)/d^2

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maxima [A]  time = 1.40, size = 33, normalized size = 0.79 \begin {gather*} \frac {2 \, {\left (7 \, {\left (d x + c\right )}^{\frac {9}{2}} b - 9 \, {\left (b c - a d\right )} {\left (d x + c\right )}^{\frac {7}{2}}\right )}}{63 \, d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c)^(5/2),x, algorithm="maxima")

[Out]

2/63*(7*(d*x + c)^(9/2)*b - 9*(b*c - a*d)*(d*x + c)^(7/2))/d^2

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mupad [B]  time = 0.05, size = 29, normalized size = 0.69 \begin {gather*} \frac {2\,{\left (c+d\,x\right )}^{7/2}\,\left (9\,a\,d-9\,b\,c+7\,b\,\left (c+d\,x\right )\right )}{63\,d^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)*(c + d*x)^(5/2),x)

[Out]

(2*(c + d*x)^(7/2)*(9*a*d - 9*b*c + 7*b*(c + d*x)))/(63*d^2)

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sympy [A]  time = 2.36, size = 194, normalized size = 4.62 \begin {gather*} \begin {cases} \frac {2 a c^{3} \sqrt {c + d x}}{7 d} + \frac {6 a c^{2} x \sqrt {c + d x}}{7} + \frac {6 a c d x^{2} \sqrt {c + d x}}{7} + \frac {2 a d^{2} x^{3} \sqrt {c + d x}}{7} - \frac {4 b c^{4} \sqrt {c + d x}}{63 d^{2}} + \frac {2 b c^{3} x \sqrt {c + d x}}{63 d} + \frac {10 b c^{2} x^{2} \sqrt {c + d x}}{21} + \frac {38 b c d x^{3} \sqrt {c + d x}}{63} + \frac {2 b d^{2} x^{4} \sqrt {c + d x}}{9} & \text {for}\: d \neq 0 \\c^{\frac {5}{2}} \left (a x + \frac {b x^{2}}{2}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)*(d*x+c)**(5/2),x)

[Out]

Piecewise((2*a*c**3*sqrt(c + d*x)/(7*d) + 6*a*c**2*x*sqrt(c + d*x)/7 + 6*a*c*d*x**2*sqrt(c + d*x)/7 + 2*a*d**2
*x**3*sqrt(c + d*x)/7 - 4*b*c**4*sqrt(c + d*x)/(63*d**2) + 2*b*c**3*x*sqrt(c + d*x)/(63*d) + 10*b*c**2*x**2*sq
rt(c + d*x)/21 + 38*b*c*d*x**3*sqrt(c + d*x)/63 + 2*b*d**2*x**4*sqrt(c + d*x)/9, Ne(d, 0)), (c**(5/2)*(a*x + b
*x**2/2), True))

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